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Directional-Amplitude Geometry of Spin-½ and Linear Photon Polarization: Deterministic Filtering, Uniqueness, and Consistency

Jorgen Karlsen

Published Oct 23, 2026• 0 citations
Publisher

Abstract

We develop a unified geometric representation of spin-½ Stern–Gerlach measurement and linear photon polarization using nonnegative distributions of individual directional amplitudes with deterministic binary responses. For spin-½, hemispherical classification of individual orientations reproduces the complete rotational transition law, P₊(α) = cos²(α/2). For linear photon polarization, an unoriented transverse axis and deterministic ±45° acceptance sectors reproduce Malus' law, P(β) = cos²β, through an exact angular overlap integral. The principal mathematical results are two support-constrained uniqueness theorems. In both geometries, requiring a normalized nonnegative distribution confined to the preparation acceptance region, together with agreement with the complete family of rotational transition probabilities, uniquely determines the corresponding cosine-weighted prepared-state density. The spherical and planar proofs establish how geometric support conditions eliminate ambiguities otherwise left unresolved by binary filtering. A systematic consistency analysis examines pristine populations, complete transition matrices, analyzer reversal, rotational covariance, immediate repeatability, operational isotropy of equal mixtures, and sequential re-preparation. The constructions reproduce the standard ideal measurement statistics while distinguishing definite individual directional amplitudes from their statistical distributions across prepared ensembles. Together, these results establish a common mathematical framework for two physically different binary measurement geometries, identifying the positive directional distributions compatible with their complete quantum transition statistics and providing a basis for investigating higher-spin measurement structures.

DOI
10.5281/zenodo.23603944

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A unifying taxonomy must explain both shared structure and physical differences. We test a common disclosure requirement: rotating a preparation and its analyzer together should preserve the probabilities. The spinor and vector actions of one Clifford rotor connect spin projection, helicity, and photon polarization while producing different rotation weights. With specified quantum measurements, we recover the angular laws and prove a precise obstruction to identifying the representations. The physical choice of representation remains an input.
Complementary Phase Geometry (CPG) contains an intrinsic three-dimensional complementary rotational sector. Here this structure is examined as a geometric carrier of spin and polarization. A normalized complementary rotation bivector defines an oriented plane whose dual axis is identified with polarization. Its spinorial lift yields the standard double-cover structure Spin(3)≃SU(2)Spin(3)\simeq SU(2)Spin(3)≃SU(2) and the half-phase relation χ2=Υ\chi^2=\Upsilonχ2=Υ. In the fundamental representation,S^a=b2σa, \hat S_a=\frac{b}{2}\sigma_a,S^a​=2b​σa​,where bbb is the CPG action scale; with the empirical identification b=ℏb=\hbarb=ℏ, the standard spin-1/21/21/2 algebra, projections, and Casimir are recovered. Combining the polarization--plane identification with the pre-existing mixed observable--complementary dynamics of CPG yields a further conditional prediction. Both the free-flight timing variance and the transverse-position variance contain the common dimensionless dependencesin⁡2ϑ sin⁡2 ⁣(ωML2v), \sin^2\vartheta\, \sin^2\!\left( \frac{\omega_{\rm M}L}{2v} \right),sin2ϑsin2(2vωM​L​),while their dimensional prefactors are different. When the rapid propagation phase is unresolved, the common angular dependence reduces to sin⁡2ϑ\sin^2\varthetasin2ϑ. The absolute magnitudes remain controlled by presently undetermined complementary and mixed-coupling parameters. Phenomenological magnetic coupling reproduces standard Stern--Gerlach eigenchannels and analyzer-overlap weights. Magnetic moments, microscopic branch selection, and Born-rule statistics remain open.
Malus’s law and the correlations of Bell-type experiments yield the same curve, the squared cosine of the angular offset between two polarizers. We propose a reinterpretation of both phenomena through a single mechanism: the correlation of responses at two points a and b, in series for Malus on a single photon, in parallel for Bell on two identical photons. No other situation is analyzed. The proposed resolution is strictly local, and reassigns cos²(a−b), observed in both cases, as the universal angular ratio of photons. The two experiments thus bring this correlation to light not because paired or polarized photons carry it, but because they make an otherwise invisible correlation testable. Bell’s theorem does not apply to this proposal, because cos²(a−b) is no longer read as the statistics of the responses at a and at b taken separately, but as the statistics of the responses at a and b taken jointly, independently of every other point. In other words, the cos² distribution of responses around an angle a holds only in association with the considered response at a, and does not constrain what the individual statistics of the response at b must be when the response at a is unknown. The article concludes that a strictly local explanation of Bell is possible again, and offers a falsifiable prediction: if identity sharing exists, understanding what it consists of will make it possible to produce identical photons without a common source.